A Bayes factor (BF) is a statistical index that quantifies the evidence for a hypothesis, compared to an alternative hypothesis (for introductions to Bayes factors, see here, here or here).

Although the BF is a continuous measure of evidence, humans love verbal labels, categories, and benchmarks. Labels give interpretations of the objective index – and that is both the good and the bad about labels. The good thing is that these labels can facilitate communication (but see @richardmorey), and people just crave for verbal interpretations to guide their understanding of those “boring” raw numbers.

The bad thing about labels is that an interpretation should always be context dependent (Such as “30 min.” can be both a long time (train delay) or a short time (concert), as @CaAl said). But once a categorical system has been established, it’s no longer context dependent.

These labels can also be a dangerous tool, as they implicitly introduce cutoff values (“Hey, the BF jumped over the boundary of 3. It’s not anecdotal any more, it’s moderate evidence!”). But we do not want another sacred .05 criterion!; see also Andrew Gelman’s blog post and its critical comments. The strength of the BF is precisely its non-binary nature.

Several labels for paraphrasing the size of a BF have been suggested. The most common system seems to be the suggestion of Harold Jeffreys (1961):

Bayes factor

Label

> 100

Extreme evidence for H1

30 – 100

Very strong evidence for H1

10 – 30

Strong evidence for H1

3 – 10

Moderate evidence for H1

1 – 3

Anecdotal evidence for H1

1

No evidence

1/3 – 1

Anecdotal evidence for H0

1/3 – 1/10

Moderate evidence for H0

1/10 – 1/30

Strong evidence for H0

1/30 – 1/100

Very strong evidence for H0

< 1/100

Extreme evidence for H0

Note: The original label for 3 < BF < 10 was “substantial evidence”. Lee and Wagenmakers (2013) changed it to “moderate”, as “substantial” already sounds too decisive. “Anecdotal” formerly was known as “Barely worth mentioning”.

Kass and Raftery suggested a comparable classification, only that the “strong evidence” category for them starts at BF > 20 (see also Wikipedia entry).

Getting a feeling for Bayes factors

How much is a of 3.7? It indicates that data occured 3.7x more likely under than under , given the priors assumed in the model. Is that a lot of evidence for ? Or not?

Following Table 1, it can be labeled “moderate evidence” for an effect – whatever that means.

Some have argued that strong evidence, such as BFs > 10, are quite evident from eyeballing only:

“If your result needs a statistician then you should design a better experiment.” (attributed to Ernest Rutherford)

Is that really the case? Can we just “see” it when there is an effect?

Let’s approach the topic a bit more experientially. What does such a BF look like, visually? We take the good old urn model as a first example.

Visualizing Bayes factors for proportions

Imagine the following scenario: When I give a present to my two boys (4 and 6 years old), it is not so important what it is. The most important thing is: “Is it fair?”. (And my boys are very sensitive detectors of unfairness).

Imagine you have bags with red and blue marbles. Obviously, the blue marbles are much better, so it is key to make sure that in each bag there is an equal number of red and blue marbles. Hence, for our familial harmony I should check whether reds and blues are distributed evenly or not. In statistical terms: : p = 0.5, : p != 0.5.

When drawing samples from the bags, the strongest evidence for an even distribution () is given when exactly the same number of red and blue marbles has been drawn. How much evidence for is it when I draw n=2, 1 red/1 blue? The answer is in Figure 1, upper table, first row: The is 0.86 in favor of , resp. a of 1.16 in favor of – i.e., anecdotal evidence for an equal distribution.

You can get these values easily with the famous BayesFactor package for R:

proportionBF(y=1, N=2, p=0.5)

What if I had drawn two reds instead? Then the BF would be 1.14 in favor of (see Figure 1, lower table, row 1).

proportionBF(y=2, N=2, p=0.5)

Obviously, with small sample sizes it’s not possible to generate strong evidence, neither for nor for . You need a minimal sample size to leave the region of “anecdotal evidence”. Figure 1 shows some examples how the BF gets more extreme with increasing sample size.

Figure 1.

These visualizations indeed seem to indicate that for simple designs such as the urn model you do not really need a statistical test if your BF is > 10. You can just see it from looking at the data (although the “obviousness” is more pronounced for large BFs in small sample sizes).

Maximal and minimal Bayes factors for a certain sample size

The dotted lines in Figure 2 show the maximal and the minimal BF that can be obtained for a given number of drawn marbles. The minimum BF is obtained when the sample is maximally consistent with (i.e. when exactly the same number of red and blue marbles has been drawn), the maximal BF is obtained when only marbles from one color are drawn.

Figure 2: Maximal and minimal BF for a certain sample size.

Figure 2 highlights two features:

If you have few data points, you cannot have strong evidence, neither for nor for .

It is much easier to get strong evidence for than for . This property depends somewhat on the choice of the prior distribution of effect sizes. If you expect very strong effects under the , it is easier to get evidence for. But still, with every reasonable prior distribution, it is easier to gather evidence for .

Get a feeling yourself!

Here’s a shiny widget that let’s you draw marbles from the urn. Monitor how the BF evolves as you sequentially add marbles to your sample!

Teaching sequential sampling and Bayes factors

When I teach sequential sampling and Bayes factors, I bring an actual bag with marbles (or candies of two colors).

In my typical setup I ask some volunteers to test whether the same amount of both colors is in the bag. (The bag of course has a cover so that they don’t see the marbles). They may sample as many marbles as they want, but each marble costs them 10 Cent (i.e., an efficiency criterium: Sample as much as necessary, but not too much!). They should think aloud, about when they have a first hunch, and when they are relatively sure about the presence or absence of an effect. I use a color mixture of 2:1 – in my experience this give a good chance to detect the difference, but it’s not too obvious (some teams stop sampling and conclude “no difference”).

This exercise typically reveals following insights (hopefully!)

By intuition, humans sample sequentially. When the evidence is not strong enough, more data is sampled, until they are sure enough about the (un)fairness of the distribution.

Intuitionally, nobody does a fixed-n design with a-priori power analysis.

Often, they stop quite soon, in the range of “anecdotal evidence”. It’s also my own impression: BFs that are still in the “anecdotal” range already look quite conclusive for everyday hypothesis testing (e.g., a 2 vs. 9 distribution; = 2.7). This might change, however, if in the scenario a wrong decision is associated with higher costs. Next time, I will try a scenario of prescription drugs which have potentially severe side effects.

The “interocular traumatic test”

The analysis so far seems to support the “interocular traumatic test”: “when the data are so compelling that conclusion hits you straight between the eyes” (attributed to Joseph Berkson; quoted from Wagenmakers, Verhagen, & Ly, 2014).

But the authors go on and quote Edwards et al. (1963, p. 217), who said: “…the enthusiast’s interocular trauma may be the skeptic’s random error. A little arithmetic to verify the extent of the trauma can yield great peace of mind for little cost.”.

In the next visualization we will see, that large Bayes factors are not always obvious.

Visualizing Bayes factors for group differences

What happens if we switch to group differences? European women have on average a self-reported height of 165.8 cm, European males of 177.9 cm – difference: 12.1 cm, pooled standard deviation is around 7 cm. (Source: European Community Household Panel; see Garcia, J., & Quintana-Domeque, C., 2007; based on ~50,000 participants born between 1970 and 1980). This translates to a Cohen’s d of 1.72.

Unfortunately, this source only contains self-reported heights, which can be subject to biases (males over-report their height on average). But it was the only source I found which also contains the standard deviations within sex. However, Meyer et al (2001) report a similar effect size of d = 1.8 for objectively measured heights.

Now look at this plot. Would you say the blue lines are obviously higher than the red ones?

I couldn’t say for sure. But the is 14.54, a “strong” evidence!

If we sort the lines by height the effect is more visible:

… and alternatively, we can plot the distributions of males’ and females’ heights:

Can we get a feeling for Bayes factors?

To summarize: Whether a strong evidence “hits you between the eyes” depends on many things – the kind of test, the kind of visualization, the sample size. Sometimes a BF of 2.5 seems obvious, and sometimes it is hard to spot a BF>100 by eyeballing only. Overall, I’m glad that we have a numeric measure of strength of evidence and do not have to rely on eyeballing only.

In a recent lecture I talked about the replication crisis in psychology. After the lecture my students asked: “We learn so much stuff in our lectures, and now you tell us that a considerable proportion of these ‘facts’ probably are just false positives, or highly exaggerated? Then, what can we believe at all?”. A short discussion soon led to the crucial question:

In the era of #repligate: What are valid cues for the trustworthiness of a study?

Of course the best way to judge a study’s quality would be to read the paper thoroughly, making an informed judgement about the internal and statistical validity, invest some extra time into a literature review, and maybe take a look at the raw data, if available. However, such an investment is not possible in all scenarios.

Here, I will only focus on cues that are easy and fast to retrieve.

As a conceptual framework we can use the lens model (Brunswick, 1956), which differentiates the concepts of cue usage and cue validity. We use some information as a manifest cue for a latent variable (“cue utilization”). But only some cues are also valid indicators (“cue validity”). Valid cues correlate with the latent variable, invalid cues have no correlation. Sometimes, there exist valid cues which we don’t use, and sometimes we use cues that are not valid. Of course, each of the following cues can be critiziced, and you certainly can give many examples where each cue breaks down. Furthermore, the absence of a positive cue (e.g., if a study has not been pre-registered, which was uncommon until recently) does not necessarily indicate the untrustworthiness.

But this is the nature of cues – they are not perfect, and only work on average.

Valid cues for trustworthiness of a single study:

Pre-registration. This might be one of the strongest cues for trustworthiness. Pre-registration makes p-hacking and HARKing unlikely (Wagenmakers, Wetzels, Borsboom, Maas, & Kievit, 2012), and takes care for a sufficient amount of statistical power (At least, some sort of sample size planning has been done. Of course, this depends on the correctness of the a-priori effect size estimate).

Sample size / Statistical Power. Larger samples mean higher power, higher precision, and less false positives (Bakker, van Dijk, & Wicherts, 2012; Maxwell, Kelley, & Rausch, 2008; Schönbrodt & Perugini, 2013). Of course sample size alone is not a panacea. As always, the garbage in/garbage out principle holds, and a well designed lab study with n=40 can be much more trustworthy than a sloppy mTurk study with n=800. But all other things being equal, I put more trust in larger studies.

Independent high-power replications. If a study has been independently replicated from another lab with high power and preferably pre-registered, this probably is the strongest evidence for the trustworthiness of a study (How to conduct a replication? See the Replication Recipe by Brandt et al., 2014).

I guess that studies with Open Data and Open Material have a higher replication rate

“Willingness to Share Research Data Is Related to the Strength of the Evidence and the Quality of Reporting of Statistical Results” (Wicherts, Bakker, & Molenaar, 2011) —> this is not exactly Open Data, because here authors only shared data upon request (or not). But it points into the same direction.

Beyond publishing Open Data at all, the neatness of the data set and the quality of the analysis script is an indicator (see also comment by Richard Morey). The journal “Quarterly Journal of Political Science” demands to publish raw data and analysis code that generates all the results reported in the paper. Of these submissions, 54% “had results in the paper that differed from those generated by the author’s own code”! My fear is that analytical code that has not been refined and polished for publishing contains even more errors (not to speak of unreproducible point-and-click analyses). Therefore, a well prepared data set and analysis code should be a valid indicator.

Open Material could be an indicator that people are not afraid of replications and further scrutiny

An abstract with reasonable conclusions that stick close to the data – see also below: “Red flags”. This includes visible efforts of the authors to explain how they could be wrong and what precautions were/were not taken.

A sensitivity analysis, which shows that conclusions do not depend on specific analytical choices. For Bayesian analyses this means to explore how the conclusions depend on the choice of the prior. But you could also show how your results change when you do not exlude the outliers, or do not apply that debatable transformation to your data (see also comment)

These cues might be a feature of a specific study. Beyond that, these cues could also be used as indicators of an authors’ general approach to science (e.g., Does s/he in general embrace open practices and care about the replicability of his or her research? Does the author have a good replication record?). So the author’s open science reputation could be another valid indicator, and could be useful for hiring or tenure decisions.

(As a side note: I am not so interested in creating another formalized author index “The super-objective-h-index-extending-altmetric-open-science-author-index!”. But when I reflect about how I judge the trustworthiness of a study, I indeed take into account the open science reputation an author has).

Valid cues for trustworthiness of a research programme/ multiple studies:

Valid cues for UNtrustworthiness of a single study/ red flags:

In a comment below, Dr. R introduced the idea of “red flags”, which I really like. These red flags aren’t a prove of the untrustworthiness of a study – but definitely a warning sign to look closer and to be more sceptical.

Most p values are in the range of .03 – .05 (or, equivalently, most t-values in the 2-3 range, or most F-values are in the 4-9 range; see comment by Dr. R below).

How does a distribution of p values look like when there’s an effect? See Daniël Lakens blog. With large samples, p-values just below .05 even indicate support for the null!

It’s a highly cited result, but no direct replications have been published so far. That could be an indicator that many unsuccessful replication attempts went into the file-drawer (see comment by Ruben below).

Too good to be true: If several low-power studies are combined in a paper, it can be very unlikely that all of them produce significant results. The “Test of ExcessSignificance” has been used to formally test for “too many significant results”. Although this formal test has been criticized (e.g., see The Etz-Files, and especially the long thread of comments, or this special issue on the test), I still think excess significance can be used as a red flag indicator to look closer.

Possibly invalid cues (cues which are often used, but only seemingly are indicators for a study’s trustworthiness):

The journal’s impact factor. Impact factors correlate with retractions (Fang & Casadevall, 2011), but do not correlate with a single paper’s citation count (see here).

I’m not really sure whether that is a valid or invalid cue for a study’s quality. The higher retraction rate might due to the stronger public interest and a tougher post-publication review of papers in high-impact journals. The IF seems not to be predictive of a single paper’s citation count; but I’m not sure either whether the citation count is an index of a study’s quality. Furthermore, “Impact factors should have no place in grant-giving, tenure or appointment committees.” (ibid.), see also a reccent article by @deevybee in Times Higher Education.

On the other hand, the current replicability estimate of a full volume of JPSP is only at 20-30% (see Reproducibility Project: Psychology). A weak performance for one of our “best journals”.

The author’s publication record in high-impact journals or h-index. This might be a less valid cue as expected, or even an invalid cue.

Meta-analyses. Garbage-in, garbage-out: Meta-analyses of a biased literature produce biased results. Typical correction methods do not work well. When looking at meta-analyses, at least one has to check whether and how it was corrected for publication bias.

This list of cues was compiled in a collaborative effort. Some of them have empirical support; others are only a personal hunch.

So, if my students ask me again “What studies can we trust at all?”, I would say something like:

“If a study has a large sample size, Open Data, and maybe even has been pre-registered, I would put quite some trust into the results. If the study has been independently replicated, even better. In contrast to common practice, I do not care so much whether this paper has been published in a high-impact journal or whether the author has a long publication record. The next step, of course, is: Read the paper, and judge it’s validity and the quality of its arguments!”

What are your cues or tips for students?

This list certainly is not complete, and I would be interested in your ideas, additions, and links to relevant literature!